A010683 -id:A010683 - cp's OEIS frontend

A114709 Triangle read by rows: T(n,k) is the number of hill-free Schroeder paths of length 2n that have k horizontal steps on the x-axis (0<=k<=n). A Schroeder path of length 2n is a lattice path from (0,0) to (2n,0) consisting of U=(1,1), D=(1,-1) and H=(2,0) steps and never going below the x-axis. A hill is a peak at height 1.

1, 0, 1, 2, 0, 1, 6, 4, 0, 1, 26, 12, 6, 0, 1, 114, 56, 18, 8, 0, 1, 526, 252, 90, 24, 10, 0, 1, 2502, 1192, 414, 128, 30, 12, 0, 1, 12194, 5772, 2006, 600, 170, 36, 14, 0, 1, 60570, 28536, 9882, 2976, 810, 216, 42, 16, 0, 1, 305526, 143388, 49554, 14904, 4110, 1044

Offset: 0

Emeric Deutsch, Dec 26 2005

Row sums are the little Schroeder numbers (A001003). Column 0 is A114710. Sum_{k=0..n} k*T(n,k) = A010683(n-1).

Riordan array ((1+3*x-sqrt(1-6*x+x^2))/(2*x*(2*x+3)),(1+3*x-sqrt(1-6*x+x^2))/(2*(2*x+3))), inverse of the Riordan array ((1-3*x)/((1-x)*(1-2*x)), x*(1-3*x)/((1-x)*(1-2*x))). - Paul Barry, Mar 01 2011

			T(4,2)=6 because we have (HH)UHD,(HH)UUDD,(H)UHD(H),(H)UUDD(H),UHD(HH) and UUDD(HH), where U=(1,1), D=(1,-1) and H=(2,0) (the H's on the x-axis are shown between parentheses).
Triangle starts:
  1;
  0,1;
  2,0,1;
  6,4,0,1;
  26,12,6,0,1;
Production matrix is
  0, 1,
  2, 0, 1,
  6, 2, 0, 1,
  18, 6, 2, 0, 1,
  54, 18, 6, 2, 0, 1,
  162, 54, 18, 6, 2, 0, 1,
  486, 162, 54, 18, 6, 2, 0, 1,
  1458, 486, 162, 54, 18, 6, 2, 0, 1
where the columns have generator (1-x)*(1-2*x)/(1-3*x).

Michael De Vlieger, Table of n, a(n) for n = 0..11475 (assuming the Kruchinin formula, rows 0 <= n <= 150, flattened.)
Emeric Deutsch, L. Ferrari, and S. Rinaldi, Production Matrices, Advances in Applied Mathematics, 34 (2005) pp. 101-122.
Shishuo Fu and Yaling Wang, Bijective recurrences concerning two Schröder triangles, arXiv:1908.03912 [math.CO], 2019.

Cf. A001003, A114710, A010683.

R:=(1-z-sqrt(1-6*z+z^2))/2/z: G:=1/(1+z-t*z-z*R): Gser:=simplify(series(G,z=0,14)): P[0]:=1: for n from 1 to 10 do P[n]:=coeff(Gser,z^n) od: for n from 0 to 10 do seq(coeff(t*P[n],t^j),j=1..n+1) od; # yields sequence in triangular form

Table[Sum[(i + 1) Binomial[k + i + 1, k] Sum[(-1)^(j + i)*2^(n - k - j - i)* Binomial[n + 1, j] Binomial[2 n - k - j - i, n], {j, 0, n - k - i}], {i, 0, n - k}]/(n + 1), {n, 0, 10}, {k, 0, n}] // Flatten (* Michael De Vlieger, Oct 30 2019 *)

T(n,k):=sum((i+1)*binomial(k+i+1,k)*sum((-1)^(j+i)*2^(n-k-j-i)*binomial(n+1,j)*binomial(2*n-k-j-i,n),j,0,n-k-i),i,0,n-k)/(n+1); /* Vladimir Kruchinin, Feb 29 2016 */

G.f.: 1/(1+z-t*z-z*R), where R=(1-z-sqrt(1-6*z+z^2))/(2*z) is the g.f. of the large Schroeder numbers (A006318).

T(n,k) = Sum_{i=0..n-k}((i+1)*binomial(k+i+1,k)*Sum_{j=0..n-k-i}((-1)^(j+i)*2^(n-k-j-i)*binomial(n+1,j)*binomial(2*n-k-j-i,n)))/(n+1). - Vladimir Kruchinin, Feb 29 2016

A378668 G.f. A(x) satisfies A(x) = 1/( 1 - xA(x)^2/(1 - xA(x)^2) )^2.

Original entry on oeis.org

1, 2, 13, 112, 1104, 11778, 132374, 1543740, 18505996, 226632616, 2823110349, 35659080952, 455652487060, 5879489288828, 76502741016012, 1002670573618324, 13224761472453756, 175404372357915096, 2338003752387818372, 31302169754776944512, 420760252068869028028

Offset: 0

Seiichi Manyama, Dec 02 2024

nonn

Cf. A243667, A378610, A378669.
Cf. A010683, A211789, A378670.
Cf. A378613.

a(n) = 2*sum(k=0, n, 2^k*(-1)^(n-k)*binomial(n, k)*binomial(4*n+k+2, n)/(4*n+k+2));

a(n, r=2, s=1, t=5, u=4) = r*sum(k=0, n, binomial(t*k+u*(n-k)+r, k)*binomial(n+(s-1)*k-1, n-k)/(t*k+u*(n-k)+r));

G.f.: exp( 1/2 * Sum_{k>=1} A378613(k) * x^k/k ).
G.f.: B(x)^2 where B(x) is the g.f. of A243667.
a(n) = 2 * Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(n,k) * binomial(4*n+k+2,n)/(4*n+k+2).
a(n) = 2 * Sum_{k=0..n} binomial(4*n+k+2,k) * binomial(n-1,n-k)/(4*n+k+2).
G.f. A(x) satisfies A(x) = ( 1 + x*A(x)^(5/2)/(1 - x*A(x)^2) )^2.

A261207 Expansion of (x-1)/8 - (x^2-4x-1)/(8sqrt(x^2-6*x+1)).

Original entry on oeis.org

0, 1, 3, 14, 70, 363, 1925, 10364, 56412, 309605, 1710247, 9496746, 52960674, 296408847, 1663998345, 9365980152, 52837614456, 298676661129, 1691325089867, 9592607927750, 54482777049918, 309837754937843, 1764046900535053, 10054065679046004, 57357471874390100

Offset: 0

Vladimir Kruchinin, Aug 11 2015

nonn

Number of vertices in all Schroeder trees with n leaves. See Theorem 2.1 of Van Duzer. - Michel Marcus, Apr 12 2019

G. C. Greubel, Table of n, a(n) for n = 0..1000
Anthony Van Duzer, Subtrees of a Given size in Schroeder Trees, arXiv:1904.05525 [math.CO], 2019.

Cf. A010683.

a := n -> simplify((-1)^(n+1)*(n*(n+1)/2)*hypergeom([1-n, 1+n], [3], 2));
seq(a(n),n=0..27); # Peter Luschny, Aug 12 2015

CoefficientList[Series[(x - 1) / 8 - (x^2 - 4 x - 1) / (8 Sqrt[x^2 - 6 x + 1]), {x, 0, 33}], x] (* Vincenzo Librandi, Aug 12 2015 *)

a(n):=sum(2^i*(-1)^(n-i-1)*binomial(n+1,n-i-1)*binomial(n+i,n),i,0,n-1);

vector(30, n, n--; sum(i=0,n-1,2^i*(-1)^(n-i-1)*binomial(n+1,n-i-1)*binomial(n+i,n))) \\ Michel Marcus, Aug 12 2015

a(n) = Sum_{i=0..n-1}(2^i*(-1)^(n-i-1)*C(n+1,n-i-1)*C(n+i,n)).
a(n) = (-1)^(n+1)*(n*(n+1)/2)*hypergeom([1-n, 1+n], [3], 2). - Peter Luschny, Aug 12 2015
a(n) = A010683(n-1)*(n+1)/2. - Peter Luschny, Aug 12 2015
a(n) ~ (3+2*sqrt(2))^n / (2^(9/4)*sqrt(Pi*n)). - Vaclav Kotesovec, Aug 17 2015
D-finite with recurrence: n*a(n) +(-2*n-5)*a(n-1) +3*(-8*n+21)*a(n-2) +(10*n-39)*a(n-3) +(-n+5)*a(n-4)=0. - R. J. Mathar, Jan 25 2020

A378670 G.f. A(x) satisfies A(x) = 1/( 1 - xA(x)^(3/2)/(1 - xA(x)^(3/2)) )^2.

Original entry on oeis.org

1, 2, 11, 78, 627, 5432, 49464, 466726, 4522871, 44747874, 450127999, 4589821576, 47333631828, 492836382192, 5173697858508, 54700317431958, 581946708333055, 6225343630256678, 66921440314606905, 722546760572660030, 7832054418695360555, 85198490262065775840

Offset: 0

Seiichi Manyama, Dec 02 2024

nonn

Cf. A243659, A369012.
Cf. A010683, A211789, A378668.
Cf. A378612.

a(n) = 2*sum(k=0, n, 2^k*(-1)^(n-k)*binomial(n, k)*binomial(3*n+k+2, n)/(3*n+k+2));

a(n, r=2, s=1, t=4, u=3) = r*sum(k=0, n, binomial(t*k+u*(n-k)+r, k)*binomial(n+(s-1)*k-1, n-k)/(t*k+u*(n-k)+r));

G.f.: exp( 2/3 * Sum_{k>=1} A378612(k) * x^k/k ).
G.f.: B(x)^2 where B(x) is the g.f. of A243659.
a(n) = 2 * Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(n,k) * binomial(3*n+k+2,n)/(3*n+k+2).
a(n) = 2 * Sum_{k=0..n} binomial(3*n+k+2,k) * binomial(n-1,n-k)/(3*n+k+2).
G.f. A(x) satisfies A(x) = ( 1 + x*A(x)^2/(1 - x*A(x)^(3/2)) )^2.

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A378668 G.f. A(x) satisfies A(x) = 1/( 1 - x*A(x)^2/(1 - x*A(x)^2) )^2.

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A261207 Expansion of (x-1)/8 - (x^2-4*x-1)/(8*sqrt(x^2-6*x+1)).

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A378670 G.f. A(x) satisfies A(x) = 1/( 1 - x*A(x)^(3/2)/(1 - x*A(x)^(3/2)) )^2.

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A378668 G.f. A(x) satisfies A(x) = 1/( 1 - xA(x)^2/(1 - xA(x)^2) )^2.

A261207 Expansion of (x-1)/8 - (x^2-4x-1)/(8sqrt(x^2-6*x+1)).

A378670 G.f. A(x) satisfies A(x) = 1/( 1 - xA(x)^(3/2)/(1 - xA(x)^(3/2)) )^2.